Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
ty1

A mass–spring–damper is described by y″ + 3y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3y′ + 9y = 0
ζ =
2.Easy
ty1

A mass–spring–damper is described by y″ + 10y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 10y′ + 100y = 0
ζ =
3.Medium
t (min)T (°C)41Ta​84 °C

A forged steel part at 460 °C is left to cool in a workshop at 28 °C. After 41 min its temperature is 84 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 460 °CTa = 28 °CT(41) = 84 °C
k = min⁻¹
4.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 1. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 1
y(1.5) =
5.Hard
tyK = 5y(0.5) = ?

A system obeys y″ + 4y′ + 8y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 4y′ + 8y = 40y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
tyy(0) = 10y(0.75) = ?

Solve y″ + 3y′ + 27.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 3y′ + 27.25y = 0y(0) = 10y′(0) = 0
y(0.75) =

Answer key

  1. 1. ζ = 0.5
  2. 2. ζ = 0.5
  3. 3. k = 0.04983 min⁻¹
  4. 4. y(1.5) = 3.124
  5. 5. y(0.5) = 2.458
  6. 6. y(0.75) = −3.221

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